q-Dependent neutron scattering: A signature of the gap anisotropy in high-Tc superconductors

The influence of the superconducting gap anisotropy on the phonon dispersion is discussed for tetragonal and orthorhombic crystals. Superconducting order parameters with different types ofs-wave, as well asd- and (s+d)-wave anisotropies, are studied. We show that along specific directions of the Brillouin zone the lineshape (energy and linewidth) of phonons strongly depends on the anisotropy of the gap. In this context, we discuss recent experimental results performed on YBa₂Cu₃O₇ single crystals. The data can be well described in terms of an anisotropics-wave gap function obtained from the nontotally screened interaction of charge carriers with long-range optical phonons. The calculations suggest, furthermore, that it is generally insufficient to describe experiments that involve the anisotropy of the superconducting order parameter solely in terms of its symmetry. Not only the symmetry, but also the specific structure of the gap as a function of k is important. This conclusion holds for those experiments that are not restricted to the pure determination of the phase of the gap.     

Electronic and Phonon Contributions to the Pairing in the Cuprates

Thermal Difference Reflectance (TDR) Spectroscopy has been used to determinethe superconducting gap parameter for several of the superconducting cupratesover a wide range of energies, extending from the infrared (0.3 eV)to the ultraviolet (5.3 eV). A contribution to the pairing is found in eachcase from the phonons, and from an electronic excitation with energy thatranges from 1.6 eV and 2.3 eV for the different compounds attributed to thed ⁹–d ¹⁰ L charge-transfer excitation between Cu and O. In every case thereflectance ratio between the superconducting and normal state, Rs/Rnplotted as a function of photon energy can be well described using theEliashberg theory. The theory also predicted a characteristic shape for thelow energy part of such spectra due to the phonons. We report theobservation of this feature in measurements on films of Tl₂Ba₂CaCu₂O₈. Wediscuss the significance of the success of the Eliashberg theory inexplaining these results and successfully predicting new effects in thelight of the correlations that had been thought to invalidate such atheoretical approach.     

Anisotropic superconductivity in the Emery model

We have investigated anisotropic superconductivity originating from intersite pairing between holes in nearest and next nearest neighboring sites in the Emery model. Strong local Coulomb correlations among holes in copper orbitals have been taken into account within the Hubbard I approximation scheme. The superconducting transition temperature has been evaluated as a function of the hole concentration. It has been shown that with the onset of superconductivity, pairing among oxygen-like quasiparticles in the mixeds-wave+d-wave channel plays the dominating role, being replaced by pairing in the extendeds-wave channel for higher concentration of holes. Superconducting correlations are mostly effective for a rather narrow range of the model parameter values, close to values derived from band structure calculations. Therefore, the coupling betweens-wave andd-wave channels seems to be a general feature of superconductivity in CuO₂ planes.     

Existence of boths andd-wave solutions of Eliashberg equations

We solve Eliashberg equations in the case of strong anisotropic electron-phonon coupling and low energy cutoff. In a simplified model representing the Fermi Surface of (CuO₂)_n planes, we exhibit ad-wave solution as well as a generalizeds-wave one.     

q-Dependent neutron scattering: A signature of the gap anisotropy in high-T c superconductors

The influence of the superconducting gap anisotropy on the phonon dispersion is discussed for tetragonal and orthorhombic crystals. Superconducting order parameters with different types ofs-wave, as well asd- and (s+d)-wave anisotropies, are studied. We show that along specific directions of the Brillouin zone the lineshape (energy and linewidth) of phonons strongly depends on the anisotropy of the gap. In this context, we discuss recent experimental results performed on YBa₂Cu₃O₇ single crystals. The data can be well described in terms of an anisotropics-wave gap function obtained from the nontotally screened interaction of charge carriers with long-range optical phonons. The calculations suggest, furthermore, that it is generally insufficient to describe experiments that involve the anisotropy of the superconducting order parameter solely in terms of its symmetry. Not only the symmetry, but also the specific structure of the gap as a function of k is important. This conclusion holds for those experiments that are not restricted to the pure determination of the phase of the gap.     

Symmetry, mechanism of the pairing, and the isotope effect in the cuprates

This paper is concerned mainly with the problem of the origin of highT _c .s-Wave vs.d-wave pairing, the isotope effect, and the strong coupling of the carriers to low-frequency optical phonon modes are discussed. This latter strong coupling is sufficient to provide highT _c .The work of VZK was sponsored by the U.S. Office of Naval Research Contract No. N00014-92-F0006.     

Intrinsic surface depression of the order parameter under mixed (s+id)-wave pair symmetry and its effect on the critical current of high-Tc SIS Josephson junctions

An intrinsic gap depression at the superconductor-insulator interface due to the very short value of the coherence length in high-T _c superconductors [HTS] is considered in the framework of a mixed (s+id)-wave pair symmetry for the order parameter ranging from pures to pured-wave. This gap depression acts as the main physical agent causing the relevant reduction ofI _c (T)R _n (T) values with respect to BCS expectations in HTS SIS Josephson junctions. Good agreement with various experimental data is obtained with both pures-wave and pured-wave symmetries of the order parameter, but with amounts of gap depression depending on the pair symmetry adopted. Regardless of the apir symmetry considered, these results prove the importance of the surface order-parameter depression in the correct interpretation of theI _c (T)R _n (T) data in HTS SIS junctions. In the case of a planar YBCO-based junction the use of the de Gennes condition allowed us to tentatively obtain an upper limit for the amount ofd-wave present in the gap of YBCO.     

Control of electron–phonon dynamics by quantum confinement in isolated Y2SiO5:Pr nanocrystal

The fluorescence spectrum and decay of isolated Y₂SiO₅:Pr³⁺nanocrystals at the glass-air interface were investigated using the confocal fluorescence microscopy and time-resolved techniques. Intense fluorescence from the upper crystal field components of the split ¹D₂ term of Pr³⁺ was observed. The fluorescence decay is characterized by the non-exponential law t^-1/2 which is caused by the back-to-back energy exchange between Pr³⁺ electronic states and long-lived nanocrystal phonons. The observed effect is very sensitive to the cluster–substrate interaction and opens a new spectroscopic way to explore the excited states of J-manifolds of RE ions directly.     

Anisotropy of the superconducting gap in the presence of electron-phonon and Coulomb interactions

d-wave ands-wave-type solutions of the BCS gap equations are studied numerically with account of long-range and short-range interaction with optical phonon and Coulomb repulsion. It is found that for realistic values of parameters the maximal value of the gap is determined mainly by unscreened long-range interactions of charge carriers with long-waveA _lg , -andB _lg -phonons. The type of the stable solution (d-wave or s-wave) essentially depends on the strength of the Coulomb repulsion as well as on the strength of other short-range interaction potentials. The conclusion is drawn that in some members of cuprate superconductors Cooper pairs may haved- wave symmetry while in others the symmetry may be of anisotropics-wave type.     

Existence of boths andd-wave solutions of Eliashberg equations

We solve Eliashberg equations in the case of strong anisotropic electron-phonon coupling and low energy cutoff. In a simplified model representing the Fermi Surface of (CuO₂)_n planes, we exhibit ad-wave solution as well as a generalizeds-wave one.     

Resonant Raman Study of Superconducting Gap and Electron-Phonon Coupling in YbBa2Cu3O7−δ

We investigate the electronic background as well as the 02-03 mode at 330 cm ^–1 of highly doped YbBa ₂ Cu ₃ O _7– in B _1g symmetry. Above the critical temperature T._c the spectra consist of an almost constant electronic background and superimposed phononic excitations. Below T _c the superconducting gap opens and the electronic background redistributes exhibiting a 2 peak at 320 cm ^–1 . We use a model that allows us to separate the background from the phonon. In this model the phonon intensity is assigned to the coupling of the phonon to inter- and intraband electronic excitations. For excitation energies between 1.96 eV and 2.71 eV the electronic background exhibits hardly any resonance. Accordingly, the intraband contribution to the phonon intensity is not affected. In contrast, the interband contribution vanishes below T _c at 1.96 eV while it remains almost unaffected at 2.71 eV.     

s -Wave Symmetry Along the c -Axis and s + d In-plane Superconductivity in Bulk YBa2Cu4O8

To clarify the order parameter symmetry of cuprates, the magnetic penetration depth λ was measured along the crystallographic directions a, b, and c in single crystals of YBa₂Cu₄O₈ via muon spin rotation. This method is direct, bulk sensitive, and unambiguous. The temperature dependences of λ _a ⁻² and λ _b ⁻² exhibit an inflection point at low temperatures as is typical for two-gap superconductivity (TGS) with s+d-wave character in the planes. Perpendicular to the planes a pure s-wave gap is observed thereby highlighting the important role of c-axis effects. We conclude that these are generic and universal features in the bulk of cuprates.     

c-axis phonons and the enhancement of pairing correlations in high-temperature superconductors

We consider the two-dimensional Hubbard model including electron-phonon interaction. Strong local correlations (U limit) are taken into account within the mean-field approximation for auxiliary boson fields. Phonon-assisted transitions between intraand interlayer states are introduced as the source of coupling between two-dimensional CuO₂ layers. This type of processes effectively leads to the nonlinear (quadratic) interaction of intralayer electrons withc-axis phonons. We construct the Eliashberg equations for the resulting Hamiltonian and evaluate the superconducting transition temperatureT _c. Our model calculation demonstrates that a pronounced enhancement ofT _c in thed-wave channel is possible. The largest enhancement ofT _c tends to take place for small hole concentrations. This means that the coupling toc-axis phonons could compete with two-dimensional correlations responsible for the onset of antiferromagnetic order. It is remarkable that the two-dimensional features in the normal state are hardly affected by this specific interlayer interaction. Therefore,c-axis two-phonon-mediated interlayer coupling can cooperate with interlayer pair tunneling and substantially contribute to an increased pairing.     

Effects of phonons and antiferromagnetic spin fluctuations on the pairing and density of states in high-T c superconductors

We solve the Eliashberg equations for a two-dimensional, tight-binding band and anisotropic interaction due to exchange of phonons and antiferromagnetic spin fluctuations. For small band fillings, a mixture of simple and extendeds-wave pairing is stable, while for band fillings closer to half-filling thed-wave pairing state becomes stable. The density of statesN() becomes highly asymmetric in for smaller band fillings, which is an effect of particle-hole asymmetry. For thed-wave stateN() is linear in for small and exhibits a logarithmic singularity at the gap amplitude. For the mixeds-wave stateN() shows the BCS singularity at the gap edge. Antiferromagnetic spin fluctuations give rise to a pseudogap inN() for the normal state.     

Theory of Cyclotron Resonance and Magneto-Optics in n- and p-Type InMnAs in Ultra-high Magnetic Fields

We present a theory for the electronic and optical properties of n- and p-type In_1–x Mn _x As in ultra-high magnetic fields. An eight-band effective mass model based on the Pidgeon–Brown model and including the wavevector dependence of the electronic states as well as the s–d and p–d exchange interactions with Mn d-electrons is used to determine the electronic states. The optical properties such as cyclotron resonance are computed using Fermi's golden rule. Comparison of the theory with ultra-high magnetic field (>50 T) cyclotron resonance experiments shows that the electron cyclotron resonance peak shifts with Mn doping and that the shift allows one to extract the Mn-electron/hole exchange parameters, and . The hole cyclotron resonance shows multiple resonance peaks, which we attribute a heavy to heavy and light to light hole transitions.     

Symmetries of Vortex Lattice Solutions of the Bogoliubov–de Gennes Equation in a Two-Dimensional Square Lattice

This report describes symmetry properties of tetragonal vortex lattice solutions of the Bogoliubov–de Gennes equation in a two-dimensional square lattice in a uniform magnetic field. The invariance group of a tetragonal vortex lattice solution is expressed in a form of G _(l) = (e + tC _2x ) (l = 0, 2, ± 1), where tC _2x is a space rotation around the x-axis accompanied with time reversal, is a kind of fourfold rotation group, and L is the magnetic translational group of the vortex lattice state. We give a new, refined definition of local symmetric order parameters (OPs) (s-wave, d-wave, and p-wave), which have a well-defined nature such that the OP (e.g., s-wave OP) at the translated site by a lattice vector (of the vortex lattice) from a site (m, n) is expressed by the OP (e.g., s-wave) at the site (m, n) times a phase factor. Winding numbers around the origin of s-wave and d-wave OPs are obtained for four types of solutions G _(l) (l = 0, 2, ± 1). It is shown that all energy bands of quasiparticles of a vortex lattice state are doubly degenerate.     

Magnetic Specific Heat of Heavily p-Type Doped (Zn,Mn)Te:P

The magnetic contribution to the specific heat of bulk crystals of Zn_1–x Mn _x Te ( x = 0.03) heavily (up to 10¹⁹ cm^–3) p-type doped with P is studied over the temperature range 0.5–15 K and magnetic field range 0–3 T. The magnetic specific heat observed at zero magnetic field indicates that a substantial part of the magnetic ions has the degeneracy of their magnetic ground state lifted by d–d and p–d exchange interactions. The effect increases for doped and annealed samples with higher concentration of conducting holes. We have also carried out a theoretical analysis that takes into account the contributions due to small magnetic clusters, single magnetic ions in crystal field of distorted crystal lattice, and low energy excitations of the p–d exchange-coupled system of local moments and carriers.     

Superflow and the depairing critical current for spin singlet-triplet states of superfluid Fermi liquid

The unitary states of superfluid Fermi liquid with singlet D and triplet P type of pairing are investigated in the framework of the weak coupling approximation. The superflow pair-breaking critical current is calculated at zero temperature and in the Ginzburg-Landau region for various values of the respective strengths of singlet and triplet components of the pairing interaction. The dependence of the mass superflow on the a ₁ (F ₁ ^s ) Landau amplitude is determined. The mixed singlet-triplet states BS (a mixture of BW anisotropic and D-wave states) and 2DS (a mixture of 2D planar and D-wave states) are found to be stable for some region of the superfluid velocity.     

Non-linear Localized Lattice Mode Coupling Mechanism and the Pseudogap in High-temperature Superconducting Cuprates

The pseudo-gap phenomena in high-temperature cuprate superconductors is an outstanding puzzle with no consensus yet on its physical origin. A previous suggestion on the role of non-linear local lattice instability modes on the microscopic pairing mechanism in high temperature cuprate superconductors is re-examined to investigate whether unusual lattice mechanisms could cause a pseudo-gap. By assuming an electron predominantly interacting with a non-linear Q ₂ mode of the oxygen clusters in the CuO₂ planes, we show that the interaction has explicit d-wave symmetry and leads to an indirect coupling of d-wave symmetry between electrons. We show that the polaron formation by the non-linear mode can cause a pseudo-gap of d-wave symmetry before the onset of coherence in the superconducting pairing. We suggest a simple phenomenological explanation of the pseudo-gap crossover temperature and the Fermi arcs. The discussion may be relevant for the pseudo-gap in non-superconducting giant magnetoresistive manganites.     

Free Energy of a d-Wave Superconductor in the Nearly Antiferromagnetic Fermi Liquid Model

A very general Hamiltonian which describes the coupling of charge carriers via a spin susceptibility is used to derive a formula for the free energy difference between the superconducting and the normal state of a d-wave superconductor. The formula is then specialized to the nearly antiferromagnetic Fermi liquid (NAFL) model and evaluated numerically. The comparison with specific heat data measured for optimally doped YBa₂Cu₃O_6.95 reveals the necessity to include even contributions far away from the Fermi surface to the free energy. The usual restriction to a narrow shell around the Fermi surface in deriving the free energy formula is proved to be inadequate.